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Bulletin of the Moscow State Regional University (Physics and mathematics)

FEATURES OF THE COMPUTATIONAL TECHNIQUES IN THE QUANTUM FIELD THEORY

A. L. Bugrimov, Nikolai Vital’evich Zverev

Published Jan 1, 2016• 0 citations
Publisher

Abstract

The computational techniques in the quantum field theory are compared with the ordinary numerical methods in the system analysis, information management and processing.In the course of comparison, coinciding and partially coinciding computational principles and techniques in these sciences have been found.In addition, a number of features of the computational methods related only to the quantum field theory are observed.These features are noted to reflect the basic principles of the quantum field theory and the physics of elementary particles.

DOI
10.18384/2310-7251-2016-2-08-17

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The computational techniques in the quantum field theory are compared with the ordinary numerical methods in the system analysis, information management and processing.In the course of comparison, coinciding and partially coinciding computational principles and techniques in these sciences have been found.In addition, a number of features of the computational methods related only to the quantum field theory are observed.These features are noted to reflect the basic principles of the quantum field theory and the physics of elementary particles.
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We introduce the basic theory of quantization of radiation field in quantum physics and explain how it relates to the theory of recursive functions in computer science. We outline the basic differences between quantum mechanics (QM) and quantum field theory (QFT) and explain why QFT is better suited for a computational paradigm — based on algorithmic requirement, countably infinite degrees of freedom and the creation of macroscopic output objects. The quanta of the radiation field correspond to the non-negative integers and the harmonic oscillator spectra correspond to the recursive computation — with the creation and annihilation operators, respectively, playing the same role as the successor and predecessor in computability theory. Accordingly, this approach relates the classical computational model and the quantum physical model more directly than the Turing machine approach used earlier. Also, the application of Lambda calculus formalism and the associated denotational semantics (that is widely used in the classical computational paradigm involving recursive functions) for applications to computational paradigm based on quantum field theory is described. Finally, we explain where QFT and conventional paradigm depart from each other, and examine the concept of fixed points, phase transitions, programmability, emergent computation and related open problems.
In quantum many-body physics, Quantum Field Theory (QFT) provides powerful tools to describe collective particle behavior. This framework employs three crucial theoretical tools: Green's functions for characterizing particle propagation and response to perturbations, Feynman diagrams for organizing perturbative expansions of quantum processes, and renormalization for improving the convergence of diagrammatic series. A key challenge in QFT is the efficient representation of finite-temperature Green's functions, particularly at low temperatures. We develop a universal framework linking the effective degrees of freedom in these functions to the rank of Lehmann eigenspace, explaining the observed compressibility in Discrete Lehmann representation (DLR). To address numerical instability of DLR in self-consistent solvers, we introduce the symmetrized discrete Lehmann representation (SDLR), which incorporates physical symmetries while maintaining computational efficiency. Applying SDLR, we study phonon-mediated superconductivity in the uniform electron gas (UEG), focusing on the Coulomb interaction effects. Our analysis reveals that the conventional Coulomb pseudopotential μ∗\mu^*μ∗ stems from two distinct effects: modifications to the Fermi Liquid factor and alterations of effective couplings. To extend our first-principle approach to lower electron densities and real materials, vertex corrections in higher-order Feynman diagrams must be systematically evaluated. This presents a significant challenge due to the factorial growth in diagram numbers at higher orders. We address this by transforming Feynman diagrams into computational graphs, enabling efficient computation reuse across shared sub-diagrams. Furthermore, we incorporate Taylor-mode automatic differentiation to streamline the computationally intensive diagrammatic re-expansion process. This approach reduces the computational complexity from exponential to sub-exponential scale relative to the differential order, as demonstrated through calculations of the UEG effective mass ratio. This research advances computational approaches within QFT by combining theoretical physics with advanced computational methods, establishing a foundation for future interdisciplinary studies of quantum many-body systems.
The article is devoted to the actual problem of teaching methods of the subjects of the educational process of modern information and computer science. The contribution of Ukrainian scientists to the development of electronic computing machines, their programming and teaching methods, prospects for further research are briefly considered. In the natural sciences, elementary units are widely used for various measurements. In particular, Planck’s constant is the smallest portion, quantum of energy, quantum of action, which is used in quantum mechanics. It is also known from the course of physics that the smallest portion of energy in classical mechanics is the energy per one degree of freedom. Logically created mathematical models and theories describing the relevant properties of matter using the specified units. Similarly, emphasis is placed on clarifying the difference between a classical computer and quantum units of information. The concepts of bit of information, qubit and trit are revealed, and depending on the defined units of information, types of computers are distinguished: classical and quantum. The concepts of qubit and trit are considered mainly as a mathematical entity, and attention is not focused on the content of the physics of quantum objects, a comparison of the indicated units of quantum information with a traditional bit of information in the quantum world is given. A historical-genetic analysis of the registers of elementary logic systems, a theoretical study of the principles of superposition and interaction between quantum states, and experimental methods of implementing qubits and their connections have been carried out. An important field for the use of quantum computers is cryptology, for solving the problems of which Grover’s method has already been created. With Grover’s quantum algorithm, it is possible to quickly search for the necessary element in an unsorted database with 2nN (n is the length of the search space of the quantum register - the number of qubits in it; N is the size of the search space). However, although the quantum simulator gives a high probability of the correct result, experimental studies also show large discrepancies. Therefore, there is a problem for further research. Quantum computers can influence each other through the phenomenon of entanglement.